1893.] On Operators in Physical Mathematics. 109 



Effect the integration, and we obtain immediately 

 and therefore, by (13), 



(17) 



Here we see the great convenience in actual work of the condensed 

 notation. At the same time, it is desirable to expand sometimes and 

 see what the developed formula looks like. "We then take the written 

 term as a central basis, making it a factor of all the rest. Thus, 



1 , 



- H ~~ H 



30. Take r = in this, and we have 



!() = ^l-,l-l-l-...., (19) 



which is the same as (20), Part I, noting that at there is x here. 

 But of course the exponential factor is now of no service, the 

 ordinary series A, equation (1) above, being the practical formula 

 when x is small. 



Take r = ^ in (18), and we obtain 



which is the formula C, equation (3) above, the practical formula 

 when x is bigger than is suitable for rapid calculation by A. Observe 

 that these are the extreme cases, for the whole of the second line in 

 (18) goes out to make (19), and the whole of the first line, excepting 

 the first term, goes out to make (20). On the other hand, it fre- 

 quently happens that extreme cases of a generalized formula are 

 numerically uninterpretable. 



To convert (18) to the form aA + 5B algebraically, we may use the 

 exponential expansion in the form (12), but with r negatived, thus, 



e*=2~. (21) 



